Refined Upper Bounds for L(1,χ)

2025/03/08 by Ezearn, Jeffery
#11M20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2503.06210

Abstract

Let χ be a non-principal Dirichlet character of modulus q with associated L-function L(s,χ). We prove that |L(1,χ)|≤((1)/(2)+O((loglog q)/(log q)))(φ(q))/(q)log q , where φ(⋅) is Euler's phi function. This refines known bounds of the form (c+o(1))log q or (c+O((1)/(log q)))log q and is relevant for prime-rich moduli. It follows from Mertens' third theorem and the prime number theorem that inf_q>2max_{χ≠χ0 (\mod q)}(|L(1,χ)|)/(log q/loglog q)≤(1)/(2)e^-γ.

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